Prior research on students' uses of technology in the context of Euclidean geometry has suggested it can be used to support students' development of formal justifications and proofs. This study examined the ways in which students used a dynamic geometry tool, NonEuclid, as they constructed arguments about geometric objects and relationships in hyperbolic geometry. Eight students enrolled in a college geometry course participated in a task-based interview that was focused on examining properties of quadrilaterals in the Poincaré disk model. Toulmin's argumentation model was used to analyze the nature of the arguments students provided when they had access to technology while solving the problems. Three themes related to the structure of students' arguments were identified. These involved the explicitness of warrants provided, uses of technology, and types of tasks.

### Karen F. Hollebrands, AnnaMarie Conner and Ryan C. Smith

### Ricardo Nemirovsky, Molly L. Kelton and Bohdan Rhodehamel

Research in experimental and developmental psychology, cognitive science, and neuroscience suggests that tool fluency depends on the merging of perceptual and motor aspects of its use, an achievement we call *perceptuomotor integration*. We investigate the development of perceptuomotor integration and its role in mathematical thinking and learning. Just as expertise in playing a piano relies on the interanimation of finger movements and perceived sounds, we argue that mathematical expertise involves the systematic interpenetration of perceptual and motor aspects of playing *mathematical instruments*. Through 2 microethnographic case studies of visitors who engaged with an interactive mathematics exhibit in a science museum, we explore the real-time emergence of perceptuomotor integration and the ways in which it supports mathematical imagination.

### Rose Mary Zbiek

This study explored the strategies used by 13 prospective secondary school mathematics teachers to develop and validate functions as mathematical models of real-world situations. The students, enrolled in an elective mathematics course, had continuous access to curve fitters, graphing utilities, and other computing tools. The modeling approaches fell under 4 general categories of technology use, distinguished by the extent and nature of curve-fitter use and the relative dominance of mathematics versus reality affecting the development and evaluation of models. Data suggested that strategy choice was influenced by task characteristics and interactions with other student modelers. A grounded hypothesis on strategy selection and use was formulated.

### P. Holt Wilson, Hollylynne Stohl Lee and Karen F. Hollebrands

This study investigated the processes used by prospective mathematics teachers as they examined middle-school students' work solving statistical problems using a computer software program. Students' work on the tasks was captured in a videocase used by prospective teachers enrolled in a mathematics education course focused on teaching secondary mathematics with technology. The researchers developed a model for characterizing prospective teachers' attention to students' work and actions and interpretations of students' mathematical thinking. The model facilitated the identification of four categories: describing, comparing, inferring, and restructuring. Ways in which the model may be used by other researchers and implications for the design of pedagogical tasks for prospective teachers are discussed.